Congruence Postulates for Triangles

Congruence Postulates for Triangles

1. CONGRUENCE
POSTULATES
FOR
TRIANGLES
SEPOCTNOVDECJANFEBMARAPRMAYJUNJULAUG
A
B
C
D
E
F

2. CONGRUENCE POSTULATES FOR TRIANGLES
●The only way to prove that two or more triangles are congruent
is to establish the following:
○Corresponding two sides are congruent
○Corresponding two sides with included angles are
congruent
○Corresponding two angles with included side are congruent
○Corresponding side with consecutive two angles are
congruent

3. SAS (Side-Angle-Side) Congruence Postulate
●If two sides and the included
angles of one triangle are
congruent with respect to two
sides and included angle of
another triangle, then the two
triangles are congruent.
●Every SAS correspondence
means congruence.
AC≌XZ
BC≌YZ
∠C≌∠Z

4. ASA (Angle-Side-Angle) Congruence Postulate
●If in two triangles, two angles
and the included side of one are
congruent with respect to two
angles and the included side of
the other, then the two triangles
are congruent.
●Every ASA correspondence
means congruence.
BC≌EF
∠B≌∠E
∠C≌∠F

5. SSS (Side-Side-Side) Congruence Postulate
●If in two triangles, the three
sides of one are congruent to
the three sides of the other,
respectively, then the two
triangles are congruent.
●Every SSS correspondence
means congruence.

AB≌DE
BC≌EF
AC≌DF

6. SAA (Side-Angle-Angle) Congruence Postulate
●If in two triangles, two angles
and non-included side of one
are congruent to two angles and
a non-included side of the
other, respectively, then the two
triangles are congruent.
AC≌DF
∠B≌∠E
∠C≌∠F

7. SEPOCTNOVDECJANFEBMARAPRMAYJUNJULAUG
A
B
C
D
E
F

8. Name that Postulate.

9. Name that Postulate.

10. Name that Postulate.

11. Name that Postulate.

12. Name that Postulate.

13. PROVING
TRIANGLES
CONGRUENT
SEPOCTNOVDECJANFEBMARAPRMAYJUNJULAUG
A
B
C
D
E
F

14. Problem no. 1

15. Step 1: Mark the Given.

16. Step 2: Mark other implications.

17. Step 3: Choose a Method.

18. Step 4: List the Parts
STATEMENTS REASONS
1. AB ≌ CD
2. BC ≌ DA
3. AC ≌ AC

19. Step 5: List the Reasons
STATEMENTS REASONS
1. AB ≌ CD Given
2. BC ≌ DA Given
3. AC ≌ AC Reflexive Property

20. Step 6: Conclude
STATEMENTS REASONS
1. AB ≌ CD Given
2. BC ≌ DA Given
3. AC ≌ AC Reflexive Property
4. ΔABC ≌ ΔCDA SSS Congruence Postulate

22. Problem no. 2

23. Step 1: Mark the given.

24. Step 2: Mark other implications.

25. Step 3: Choose a method.

26. Step 4: List the parts.
STATEMENTS REASONS
1. AB ≌ CB
2. EB ≌ DB
3. ∠ABE ≌ ∠CBD

27. Step 5: Fill in the reasons.
STATEMENTS REASONS
1. AB ≌ CB Given
2. EB ≌ DB Given
3. ∠ABE ≌ ∠CBD Vertical Angles Theorem

28. Step 6: Conclude.
STATEMENTS REASONS
1. AB ≌ CB Given
2. EB ≌ DB Given
3. ∠ABE ≌ ∠CBD Vertical Angles Theorem
4. ΔABE ≌ ΔCBD SAS Congruence Postulate

30. Problem no. 3

31. Step 1: Mark the given.

32. Step 2: Mark other implications.

33. Step 3: Choose a method.

34. Step 4: List the parts.
STATEMENTS REASONS
1. ∠XWY ≌ ∠ZWY
2. ∠XYW ≌ ∠ZYW
3. WY ≌ WY

35. Step 5: Fill in the reasons.
STATEMENTS REASONS
1. ∠XWY ≌ ∠ZWY Given
2. ∠XYW ≌ ∠ZYW Given
3. WY ≌ WY Reflexive Property

36. Step 6: Conclude.
STATEMENTS REASONS
1. ∠XWY ≌ ∠ZWY Given
2. ∠XYW ≌ ∠ZYW Given
3. WY ≌ WY Reflexive Property
4. ΔWXY ≌ ∠ΔWZY ASA Congruence Postulate

40. Step 1: Mark the given.

41. Step 2: Mark implications.

42. Step 3: Choose a method.

43. Step 4: List the parts.
STATEMENTS REASONS
1. AB ≌ AD
2. AC bisects ∠BAD
3. AC ≌ AC
4. ∠BAC ≌ ∠DAC

44. Step 5: Fill in the reasons.
STATEMENTS REASONS
1. AB ≌ AD Given
2. AC bisects ∠BAD Given
3. AC ≌ AC Reflexive Property
4. ∠BAC ≌ ∠DAC Definition of Bisector

45. Step 5: Conclude.
STATEMENTS REASONS
1. AB ≌ AD Given
2. AC bisects ∠BAD Given
3. AC ≌ AC Reflexive Property
4. ∠BAC ≌ ∠DAC Definition of Bisector
5. ΔABC ≌ ΔADC SAS Congruence Postulate

47. STATEMENTS REASONS
1. C is the midpoint of BD Given
2. AB ≌ AD Given
3. BC ≌ CD Def. of Midpoint
4. AC ≌ AC Reflexive Property
5. ΔABC ≌ ΔADC SSS Congruence Postulate

48. STATEMENTS REASONS
1. AB || DC Given
2. ∠BAC ≌ ∠DCA Alternate Interior Angles Theorem
3. AD || BC Given
4. ∠DAC ≌ ∠BCA Alternate Interior Angles Theorem
5. AC ≌ AC Reflexive Property
5. ΔABC ≌ ΔCDA ASA Congruence Postulate

49. STATEMENTS REASONS
1. AD bisects EC Given
2. EB ≌ BC Def. of Bisector
3. EC bisects AD Given
4. AB ≌ BD Def. of Bisector
5. ∠ABE ≌ ∠BCD Vertical Angle Theorem
5. ΔABE ≌ ΔDBC SAS Congruence Postulate

50. STATEMENTS REASONS
1. AC bisects ∠BAD Given
2. ∠BAC ≌ ∠CAD Def. of Angle Bisector
3. AB ≌ BD Given
5. AC ≌ AC Reflexive Property
5. ΔABC ≌ ΔADC SAS Congruence Postulate